I am trying to understand what the significance is of the face for which a force is acting on when talking about a stress tensor.
Say we consider the components $T_{xx}$ and $T_{zx}$ of the stress tensor $T$.
Both of these refer to stress acting in the negative $x$ direction. Yet the first one is acting on the face orthogonal to the $x$ axis, and the second one is acting on the face orthogonal to the $z$ axis.
What is the significance of this? The stress tensor is defined for every point so it seems that at a particular point the stress tensor is describing two different forces acting in the same direction...I just don't really get what's going on, can someone explain it clearly?
Try to formalize what internal stress state should actually mean. What science has come up with is the notion of sectional forces and moments, together with a way to relate them to internal stress state.
You need to do a Gedankenexperiment: Imagine a smooth cut passing through the material, including the point $p$. Remove the material on one side of the cut and orient the normal vector $n$ of the cut at $p$ such that it points toward the side where material has been removed. In order to keep the mechanical state of the remaining material identical to its pre-cut version, a certain vector field of forces-per-area would have to be applied along the cut. We now suppose that this is what the now-removed material had previously exerted. At point $p$, you can sample that force-per-area vector field by computing the matrix-vector product $T\cdot n$. It really depends on how that imaginary cut passes through $p$ and which side you remove: In particular, if you remove the other side, you ought to find exactly opposing counter forces (actio = reactio), and the formalism ensures this because $T\cdot(-n) = -(T\cdot n)$. The dependency on the particulars of the cut now lies solely in $n$, and the stress tensor $T$ is by itself independent from cuts, but applicable to all ways of cutting, and therefore $T$ is eligible as a description of internal stress state.
To see why an expression of the form $T\cdot n$ has to be used, imagine a region inside the material that is enclosed by a small polyhedron. Let us name the amounts of area covered by its surface polygons $|A_1|,\ldots,|A_k|$ and the associated outward-pointing surface normal vectors $n_1,\ldots,n_k$. The fact that the polyhedron is closed is expressed with the vector equation $$\sum_{i=1}^k |A_i|\,n_i = 0$$ If the polyhedron is small enough that the enclosed mass is negligible and that the internal stress state is essentially the same near each surface polygon, the force vectors $f_1,\ldots,f_k$ acting on its faces must balance: $$\sum_{i=1}^k f_i = 0$$ To ensure this for arbitrarily shaped polyhedra, the map from $|A_i|\,n_i$ to $f_i$ must be linear. In other words, there must be a tensor $T$ such that $$f_i = T\cdot n_i\,|A_i| \quad\therefore\quad \frac{1}{|A_i|}f_i = T\cdot n_i$$ and this is what has been indicated above.
You will encounter sectional entities throughout all branches of continuum mechanics, but most importantly in the mechanics of solids. The theories of strings, rods, beams and arcs have 1D examples in the form of sectional forces $N(x)$ (longitudinal), $Q_y(x), Q_z(x)$ (transversal) and moments $M_y(x), M_z(x)$ (bending) and $M_x(x)$ (torsion), except that strings do not carry moments. Since there is only one coordinate ($x$) of interest, the normal vector $n$ reduces to the scalar factor $+1$ if you decide to cut away the part to the right, or $-1$ if you cut away the part to the left. This is again a manifestation of actio = reactio.
Theories for membranes, disks, plates and shells are mostly analogous to those for strings, rods, beams and arcs in their equations, but coordinates now span a 2D domain and e.g. for plates you will now deal with $2\times2$ matrix fields of sectional in-plane forces $N(p)$ or sectional bending moments $M(p)$ and 2D vector fields of sectional transversal forces $q(p)$. To get actual moment-per-length or force-per-length values out of these, you need to take the dot product with a 2D normal vector $n$ which describes how the imaginary cut passes through the 2D point $p$ of interest and which of its two sides is cut away.
To conclude, the 3D stress tensor is not the only mechanical entity related to the concept of sectional forces, and this emphasizes the importance of understanding the concept.